That said , the first time I had to use paper I felt like I had no idea what I was doing, because a lot of the EMR defaults aren't there to prompt you. So you have to specify a 'start time' for a medication, but in the EMR now is the default, so I didn't think about it as anything mandatory.
There are bundled minor upgrades (think service packs) that have short (10-60 minute) downtimes, blocking migrations run in that time but are kept to a minimum. If it is at all possible, migrations will be async (even on major upgrades) and run in the background after the new version goes live.
During downtime, bloodwork and the like involve pre-allocated barcode labels. Either the results will be accessible on the instrument afterwards in its memory, or the results will be printed out and it is up to the lab tech to transcribe the results.
I can't speak in depth for radiology, but I expect that they have file share servers involved for the raw results. Likely less local memory (in number of results that can be stored) than clinical lab instruments.